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This modern introduction to infinitesimal methods is a translation of the book Métodos Infinitesimais de Análise Matemática by José Sousa Pinto of the University of Aveiro, Portugal and is aimed at final year or graduate level students with a background in calculus. Surveying modern reformulations of the infinitesimal concept with a thoroughly comprehensive exposition of important and influential hyperreal numbers, the book includes previously unpublished material on the development of hyperfinite theory of Schwartz distributions and its application to generalised Fourier transforms and harmon

Nonstandard mathematical analysis. --- Calculus. --- Schwartz distributions.

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The aim of this book is to make Robinson's discovery, and some of the subsequent research, available to students with a background in undergraduate mathematics. In its various forms, the manuscript was used by the second author in several graduate courses at the University of Illinois at Urbana-Champaign. The first chapter and parts of the rest of the book can be used in an advanced undergraduate course. Research mathematicians who want a quick introduction to nonstandard analysis will also find it useful. The main addition of this book to the contributions of previous textbooks on nonstan

Mathematical analysis --- Nonstandard mathematical analysis. --- Nonstandard mathematical analysis --- Analysis, Nonstandard mathematical --- Mathematical analysis, Nonstandard --- Non-standard analysis --- Nonstandard analysis --- Model theory

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Pro matematiku dvacátého století je príznacné, že její hlavní proud nekonecno zkoumající a zároven aplikující, byt v bizarních ideálních svetech, je založen na klasické Cantorove teorii nekonecných množin. Ta sama se pak opírá o problematický predpoklad existence množiny všech prirozených císel, jehož jediné - a to navíc teologické - oduvodnení bývá zamlcováno a vytlacováno do kolektivního nevedomí.I když autor uvádí nekterá durazná varování znamenitých matematiku pred nebezpecími skrytými v soucasné infinitní matematice, není jím budovaná nová infinitní matematika jen pouhou negací soucasných názoru a predpokladu. Naopak, ta infinitní matematika, do níž predbežným úvodem je tento spisek, je vedena opatrnou snahou o nová prekracování obzoru ohranicujícího antický geometrický svet.

Nonstandard mathematical analysis. --- Analysis, Nonstandard mathematical --- Mathematical analysis, Nonstandard --- Non-standard analysis --- Nonstandard analysis --- Model theory

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Provability, Computability and Reflection

Logic, Symbolic and mathematical -- Periodicals. --- Logic, Symbolic and mathematical. --- 517.1 --- 517.1 Introduction to analysis --- Introduction to analysis --- Nonstandard mathematical analysis --- Logic, symbolic and mathematical --- Analyse mathématique non standard --- Nonstandard mathematical analysis.

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Nonstandard analysis was originally developed by Robinson to rigorously justify infinitesimals like df and dx in expressions like df/dx in Leibniz' calculus or even to justify concepts like delta-`functions'. However, the approach is much more general and was soon extended by Henson, Luxemburg and others to a useful tool especially in more advanced analysis, topology, and functional analysis. The book is an introduction with emphasis on those more advanced applications in analysis which are hardly accessible by other methods. Examples of such topics are a deeper analysis of certain functionals like Hahn-Banach limits or of finitely additive measures: From the viewpoint of classical analysis these are strange objects whose mere existence is even hard to prove. From the viewpoint of nonstandard analysis, these are rather 'explicit' objects. Formally, nonstandard analysis is an application of model theory in analysis. However, the reader of the book is not expected to have any background in model theory; instead knowledge of calculus is required and, although the book is rather self-contained, background in more advanced analysis or (elementary) topology is useful.

Nonstandard mathematical analysis --- Applied Mathematics --- Engineering & Applied Sciences --- Nonstandard mathematical analysis. --- Model theory. --- Logic, Symbolic and mathematical --- Analysis, Nonstandard mathematical --- Mathematical analysis, Nonstandard --- Non-standard analysis --- Nonstandard analysis --- Model theory --- Global analysis (Mathematics). --- Analysis. --- Analysis, Global (Mathematics) --- Differential topology --- Functions of complex variables --- Geometry, Algebraic --- Mathematical analysis. --- Analysis (Mathematics). --- 517.1 Mathematical analysis --- Mathematical analysis